Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier modulation scheme. Data is modulated into a number of separate, orthogonal signal frequencies and a portion of the data is transmitted on each of these frequencies, each known as a carrier. This scheme provides benefits over single carrier methods that are subject to interference and channel fading. OFDM transmits data through a multitude of carriers, thereby preventing channel conditions from affecting the whole signal. Additionally, the same amount of data may be transmitted at a lower symbol rate in OFDM, thereby reducing the effects of inter-symbol interference (ISI) caused by channel variations between symbols with the use of a guard interval that prevents delayed signals from interfering with a current symbol.
Referring to FIG. 1, an overview of a conventional OFDM system is shown. A coded, modulated symbol 101 is transmitted in the frequency domain. An Inverse Fast Fourier Transform IFFT 105 is used to transform the data sequence S(k) 101 of length N from the frequency domain into a time domain signal x(n) 107.
                              x          ⁡                      (            n            )                          =                              N                    ⁢                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                          S                ⁡                                  (                  k                  )                                            ⁢                              ⅇ                                  j                  ⁢                                                            2                      ⁢                                                                                          ⁢                      π                                        N                                    ⁢                  nk                                                                                        Equation        ⁢                                  ⁢                  (          1          )                    
The guard interval with Length Ng, chosen to be larger than the expected channel delay spread, is inserted into the beginning of the symbol 109. The transmitted signal xd(n) 111, will be transmitted through a linear time variant channel 113. The time variant channel 113 is modeled by the time-variant discrete impulse response h(n,l), defined as the time, n, response to an impulse applied at time n-l. Assume the maximum channel delay Nh≦N, the received signal at the receiver, yd(n) 115 may be represented as:
                                                        y              d                        ⁡                          (              n              )                                =                                                    ∑                                  l                  =                  0                                                                      N                    h                                    -                  1                                            ⁢                                                h                  ⁡                                      (                                          n                      ,                      l                                        )                                                  ⁢                                                      x                    d                                    ⁡                                      (                                          n                      -                      l                                        )                                                                        +                          w              ⁡                              (                n                )                                                    ,                                  ⁢                  0          ≤          n          ≤          N                                    Equation        ⁢                                  ⁢                  (          2          )                    
Where the w(n) 117 is the white Gaussian noise with variance σ2. After removal of the guard interval 119 at the beginning, the received signal sequence y(n) 121 will be passed to an N-points Fast Fourier Transform (FFT) 123 converting the received time domain signal 121 back to the frequency domain received signal Y(k) 125.
                              Y          ⁡                      (            k            )                          =                              1                          N                                ⁢                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                          y                ⁡                                  (                  n                  )                                            ⁢                              ⅇ                                                      -                    j                                    ⁢                                                            2                      ⁢                                                                                          ⁢                      π                                        N                                    ⁢                  nk                                                                                        Equation        ⁢                                  ⁢                  (          3          )                    
As will be explained in more detail below, the transmitted symbol x(n) 107 will be estimated from the received frequency domain signal Y(k) 125 to arrive at an estimate of the transmitted symbol X(k) 129. Using F to denote the IDF unitary transform matrix, Ht to denote the time-variant circular convolution matrix such that:Ht(n,l)=h(n,<n−l>N).
Using the defined values: x≡[x(0), x(1), . . . , x(N−1)], X≡[X(0), X(1), . . . , X(N−1)] and w≡[w(0), w(1), . . . ,2(N−1)], Equation (3) may be written as:
                                                        Y              =                            ⁢              Fy                                                                          =                            ⁢                                                FH                  t                                +                Fw                                                                                        =                            ⁢                                                                    FH                    t                                    ⁢                                      F                    H                                    ⁢                  X                                +                Fw                                                                        Equation        ⁢                                  ⁢                  (          4          )                    
Defining the sub-carrier coupling matrix:Hd=FHtFH  Equation (5)
Equation (4) could be written as:Y=HdX+ w,  Equation (6)where w is still white Gaussian noise.
With some derivation, it could be shown that Hd(k,p)=hd(k,k−p) where:
                                          h            d                    ⁡                      (                          k              ,              d                        )                          =                              1            N                    ⁢                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                          ∑                                  l                  =                  0                                                  N                                      h                    -                    1                                                              ⁢                                                h                  ⁡                                      (                                          n                      ,                      l                                        )                                                  ⁢                                  ⅇ                                                            -                      j                                        ⁢                                                                  2                        ⁢                                                                                                  ⁢                        π                                            N                                        ⁢                                          (                                              lk                        +                        dn                                            )                                                                                                                              Equation        ⁢                                  ⁢                  (          7          )                    
Notice that hd(k,0) appears on the main diagonal, hd(k,1) appears on the first sub-diagonal, hd(k,−1) appears on the first super-diagonal and so on. In more detail, the elements on the diagonal of matrix Hd, Hd(k,k), could be represented as Hd(k,k)=hd(k,k−k)=hd(k,0); the elements on the first sub-diagonal, Hd(k,k+1)=hd(k,k−(k+1))=hd(k,−1) and the elements on the first super-diagonal, Hd(k,k−1)=hd(k,k−(k−1))=hd(k,1). The relationship of hd and Hd is depicted in FIG. 2A. It is well known that hd(k,d) goes to 0 very quickly when |d| increases. Therefore, the sub-carriers coupling matrix Hd is a banded matrix with two triangles on the upper right and lower left corners, as shown in FIG. 2B. In most OFDM systems, those sub-carriers on two sides are not used (null carriers). In those cases, the two triangles in FIG. 2B could be removed, and sub-carriers coupling matrix Hd could be simplified to FIG. 2C. D=2 will be enough for most systems.
The OFDM signal may be viewed as a combination of combined sinusoidal signals when considered in the time domain. Therefore a Fast Fourier Transform (FFT) 123 may be used to convert the signal into its component frequencies on the frequency domain. When the carrier frequencies have been separated, the data may be estimated to derive the transmitted data 127.
In order to successfully isolate the carrier components, it is a basic assumption in OFDM that the carriers maintain their orthogonality. As long as the orthogonality of the carriers holds, the carriers will not interfere with each other. In a static environment, this is generally the case, but in a high mobility environment, because of the channel changing inside one OFDM symbol, the orthogonality of the carriers is not maintained and neighboring carriers begin to interfere with each other. The result is a banded carrier coupling matrix like that shown in FIG. 2B or 2C which makes the symbol estimation much more complex than a straight forward diagonal matrix as depicted in FIG. 2D. This inter-carrier interference (ICI) causes variance in the channel over a single symbol and, if not addressed, will cause an error floor. That is, the bit error rate (BER), will not improve despite a rise in the signal to noise ratio (SNR).
Compensating for ICI requires highly complex calculations, and in the case where a large number of sub-carriers are used, high dimensional arrays that require inversions are too large to be implemented in current hardware. Simple solutions that ignore ICI or only consider a nominal number of filter taps do not provide efficient and accurate estimation of the channel. It would therefore be beneficial to have an inter-carrier interference cancellation scheme that is practicable and provides an accurate estimate of the OFDM channel.